Integrating apparatus



April 9, 1946. M. M NATT 2,398,238

INTEGRATING APPARATUS Filed May 1942 1 l I 7 .22 2| n 23 ZLF 'SOALERSHARPENER IN VENTOR.

F/G. i

ATTORNEY Patented Apr. 9, 1946 INTEGRATING APPARATUS Eugene M. McNatt,Tulsa, :11., assig'nor to Standard Oil Development Company, acorporation of Delaware Application May 9, 1942, Serial No. 442,351

4 Claims.

The present invention is directed to a method and apparatus forintegrating mathematical functions.

In many physical and mathematical problems it is necessary to performrepeatedly integrations of the form white-area wherein p(a:) and q(x)may be known only as curves and 11:0 or a constant. Such integration arecommonly carried out arithmetically by plotting the two curves,measuring the value of each function at equally-spaced values of x, mul.tiplylng the values of p(:ry) and q(a:) for each measured value of m,and adding the resulting products. Since this process is laborious andtime consuming, various machines have been devised for performing thisoperation mechanically or electrically. A representative type of suchmachines is. described in The Cinema Integraph, Hazen, Brown andHedeman, Journal of the Franklin Institute, July 1940, in which may alsobe found illustrations of the type of integrals, to the solution ofwhich the present invention is directed.

The principal object of the present invention is the provision of arelatively simple method and apparatus for performing integrations ofthe character referred to with any desired degree of accuracy. Briefly,the method involves the creation of light flashes, the number of whichis 'a function of the curves represented by p(a:y)

and q(:c) and, more specifically, a function of the product of these twocurves.

The present invention may be more clearly understood from the followingdetailed description of the accompanying drawing in which Fig. 1 is aplan view of one embodiment of the present invention; I

Fig. 2 is a detail thereof;

Fig. 3 is a detailed view of a light mask used in the arrangement shownin Fig. 1; and

Fig. 4 is a similar view of a different light mask.

Briefly, the arrangement shown in Fig. 1 consists of a pair of revolvingmasks interposed between a light source and a photocell or other devicecapable of registering lighte flashes. One of the masks has drawn on ita curve represented by the equation z=p(a:) and the other has drawn onit a curved represented by the equation z=q(.'c). Both masks have the:z: axis parallel to the axis of rotation and their axes of rotation areparallel to each other and perpendicular to the beam of light. Each maskis so constructed that for each revolution it will pass a number offlashes of light which will be a function of the value of z for thevalue of a: at the point on the mask where the light beam strikes. TheR. P. M. of one mask is made a large multiple of the R. P. M. of theother mask, and upon the size of this multiple will partially depend theaccuracy of the integrations.

Referring to Fig. 1 in detail, C1 and C2 designate two hollow,transparent cylinders supported by bearings I, 2, 3 and 4. Thesebearings are the conventional ring type ball bearings. It will beobserved that the driving shafts 6 and I, respectively, for thesecylnder are journaled'in bearings I and 2, while the outer surface ofthe cylinders themselves are journaled in bearings 3 and 4'. This-isnecessary because the outer ends of the cylinders are. open. Thebearings are suitable mounted on a base plat 5. Shaft 6 is driven by asuitable motor, and is geared to shaft 1 in the manner shown. It will beapparent that any gear ratio may be employed in this transmission.

In the open ends of the cylinders, tubes 8 and 9 are mounted so as tohave their axes coincident with the axes of rotation of cylinder C2 andC2, respectively. These tubes are rigidly mounted in a block l0 which isarranged to slide on spaced tracks ll attached to the base plate 5. Theblock Ill carries a screw threaded rider l2 which is in threadedengagement with a screw l3, one

end of which is journaled in a, bearing l4 and the other end of whichterminates in a gear in the gear box in mesh with a driving gear fixedto the shaft 1. The pitch of the screw l3 can be arbitrarily selected,but th smaller it is made the greater will be the accuracy of theintegration.

Fixed in the tube 8 is a suitable light source l5, such as an automobilelamp bulb, and a mirror l6 adapted to receive a beam of light fromsource l through a pin hole I! and reflect it through a pin hole l8 inthe wall of tube 8 in a line perpendicular with the axes of rotation ofcylinders C1 and C: and in the plane thereof. Tube 9 is provided with ahole l9 to pass said beam of light. Fixed in tube 9 in register withsaid hole is a light sensitive device 20, such as a photocell, which inthe embodiment shown i connected electrically to an amplifying andsharpening cir cult 2|. A suitable sharpening circuit for this purposeis that describedby Huntoon and Strohmeyer in an article in the Reviewof Scientific 65 Instruments, 12, 35 (1941) entitled A hard vacuum tubepulse equalizing sharpening circuit. The purpose of this circuit is toconvert the pulses of electricity generated by the photocell 20 into asuitable form to operate a scaling circuit 22 to which they aredelivered. Connected to the scaling circuit is a counter 23 whichdisplays a numercial summation of the pulses delivered to it.

As shown in Fig. 2. the surface of the transparent cylinder C1 carries aseries of opaque bands 24 parallel to each other and to the axis of thecylinder. In general these bands will be very narrow and so spaced thatthe transparent and opaque strips are of equal width. A preferred methodof producing the stripes is to form them on transparent material, suchas Cellophane, with India ink and then fasten the transparent sheet onthe cylinder C1 by means of glue of transparent type. It will beobserved that only cylinder C1 is so striped.

In Figs. 3 and 4 are shown masks which may be mounted on cylinders C1and C2. For the practice of the invention it is immaterial which mask ismounted on which cylinder. In Fig. 3, for example, the mask has atransparent portion 25, the upper boundary of which follows a curvedesignated by the equation z=p(:c) in which 2 represents the ordinateand a: represents the abscissa. The remaining portion 28 of the mask isopaque. This mask is wound on the cylinder with its axis parallel to theaxis of rotation of the cylinder.

In Fig. 4 the mask illustrated has a transparent portion 21. The upperboundar is a curve represented by the equation z=q(a:). ing portion 28of this mask is opaque. It will be obvious that the portions 25 and 21of the respective masks can be cut out entirely. The masks arepreferably made by tracing the curves on transparent sheets and thenblackening the portions 26 and 28.

The masks are of such dimensions that the maximum value of z is equal tothe circumference of the corresponding cylinder, and th maximum value ofa: is not greater than the length of the corresponding cylinder. As hasbeen previ ously indicated, either mask can be placed on eithercylinder, but for the purpose of discussion it will be assumed that themask of Fig. 3 is placed on cylinder C1 and the mask of Fig. 4 is placedon cylinder C2.

It should now be clear that, as the cylinders rotate, the beam of lightpassing through C1 and C: will be chopped into flashes by the stripes onC1 and that this flashing beam will be interrupted at intervals both byC1 and C: as the opaque portions of the masks intercept the beam. Theresultant pulsating signal from the photocell is amplified and thepulses sharpened as previously described by circuit 2! and counted.

To operate the device the masks are placed on the cylinders and thetubes 8 and 9 are positioned so that the beam of light passes throughx=-u and z=0 on the mask on C1, and a:=z=0 on the mask on C2. Then, withthe amplifier and scaler ready to function, the reading of the electriccounter is noted or is set at zero. Then the motor is started andallowed to run until the screw it has advanced the beam of light to thepoint corresponding to z=t--u on the mask on cylinder Cl and :c=t on themask on C: where t is the upper limit of the integral, At this point theelectrical counter is disconnected from the the scaler, the motor isturned on, and the reading of the counter is noted.

The manner in which the machine performs The remainasoaass the desiredmultiplication and summation will now be considered. Assume that theapparatus has been started with the ray of light passing through thepoint :z:=9 and z=0 on C1 and :c=0=z on Ca. Let the total number 01clear strips on Ci be- N (disregarding the mask) and let thecircumference of C1 be D1 and the circumi'erence of C: be D2. Let thegear ratios be such that C1 rotates n revolutions to one revolution ofC2, and let the pitch of screw it be such that for one revolution of C:the beam of light advances a distance S=c.A:z: in the positive a:direction, where A: is the increment in a: corresponding to Sv and c isthe scale; that is, c is the ratio of the length of a given increment inthe :c direction to its value in the units in which a: is expressed. Forpurposes of clarity, it will be assumed in the following discussion thatput-11) and q(:::) remain constant over any given increment, As. Themaximum error .introduced by this approximation will clearly beproportional to A2.

During the first revolution of C1 the number of flashes of light, n,passing through C1 is proportional to where a is the constant ratio 01'the height of p(:r1l) as measured on C1 from the a: axis to thenumerical value of p(a:1/). In other words, it is a constant which is acharacteristic of the mask itself, and represents the scale to which thecurve on the mask is plotted.

During the different revolutions of C1, C2 rotates Dz/n times in thepositive 2 direction and the light beam advances a distance 8/11 in thepositive 1: direction.

Assuming that b times q(0) is greater than Da/n, where b is a constantfor C: corresponding to a for C1, 1 flashes of light will also passthrough C: and actuate the photocell and be registered by the sealer andcounter. Furthermore, 1 flashes of light will continue to be registeredper revolution of C1 until C: has rotated through the distance b times11(0). The number of revolutions, m1, of C1 corresponding to thisfraction of a revolution of C: is the number of revolutions of C1 perrevolution of C: times the ratio of b times qua) to the totalcircumference of C2. That is During the remaining part of the revolutionof where numerator and denominator have been multiplied by S=c.A:c.Since during subsequent revolutions of C: the above process will berepeated, we may write where the constants have been lumped into K, andwhere the subscript refers to the i revolution of C2. The grand totalnumber of flashes asaaasa of light registered as the light beam advancesfrom a:= to r=t will then be r r' EFF right-mam i t I ginrr}; pe-wqw xwhere R. is the difference between the final and initial counterreadings and d is the scaling factor. The error in the results willevidently be at least roughly proportional to Ax/nN, and it isconsequently desirable to make this quantity as small as practicable. Itis for this reason that a scaling circuit is used, for by this means amuch larger value of N can be used than would otherwise be possible toregister and still operate the machine at a reasonably fast rate.Examples of practical values of these constants for moderate accuracyare Az=1 mm., n=N= 100.

In the foregoing discussion, reference has been made to an equationcontaining the values put-y) and q(:z:) These values have beenrepresented on the masks shown in Figs. 3 and 4 as being curves of acertain shape. It will be obvious that the curve may assume any shape,depending upon the specific functions of a: represented by these values.

Other methods for generating pulses of light of a number which will be afunction of the product of the values of p(:r-y) and q(a:) and willrepresent the summation of these products for different values of a:between zero and it than that specifically disclosed will occur to thoseskilled in the art. One obvious modification is to omit the stripes fromcylinder C1 and generate the initial light flashes by inserting ashutter in the form of a toothed wheel between the.-

light source i5 and the mirror It in the manner disclosed in mycopending application Serial No. 405,910. In this case, both the shutterand the light source will be arranged outside the tube 8 so that meansfor driving the shutter at a speed which bears a fixed ratio to the R.P. M.s of the cylinders C1 and C2 can be utilized.

As has been pointed out in the foregoing description, the accuracy ofthe device can be increased by increasing the gear ratio between Cr andC2, decreasing the width of 'the stripes on C1, and decreasing the pitchof the thread on screw I3. It can be seen, therefore, that the machineis capable of considerable adjustment, depending upon the type ofproblem to be dealt with and the degree of accuracy desired.

In the above-description it has been assumed that p(a:-y)' and q(:r) areboth always positive. In the event that this is not true, separate masksmay be made for the positive and negative portions of the two functions.then be made up of four parts:

(a) the positive products corresponding to the positive portions of eachfunction;

(b) the negative products or the negative por- The integral will tion ofthe first times the positive. portion of the second;

(c) the negative products of the positive portion of the first times thenegative portion of the second; and

(d) the positive products corresponding to the negative portion or thefirst times the negative portion of the second.

It is clear that these separate parts of the integral may be obtained byfour separate integrations, and the total integral is then determined bythe algebraic sum.

The nature and objects of the present invention having been thusdescribed and illustrated, what is claimed as new and useful and isdesired to be secured by Letters Patent is:

1. An apparatus for integrating an equatiorr of the type where y is 0 ora constant and (a) and Ike) can be represented by curves, comprising alight source, a light receiver spaced from said source and in alignmenttherewith, a plurality of rotative screens interposed between said lightsource and said receiver, markings on said screens to cause the light ofsaid source to be transmitted to said receiver in a number of flashes,the numintegrating an equation where 11 is 0 or a constant and p(a:) andqua) can be represented by curves comprising a cylinder having atransparent portion and an opaque portion, a light source arranged alongy the axis of said cylinder adapted to cast a beam of light normal tothe wall thereof, a second cylinder arranged with its axis parallel tothe axis of the first cylinder and having an opaque portion and atransparent portion, the transparent portion of one of said cylindersbeing an area defined by the curve z=p(.'c) and the v transparentportion of the other being an area defined by the curve z=q(a:), the :caxis in each case being parallel with the axis of the cylinder, aphotocell arranged on the axis of said second cylinder in a position toreceive a light beam from said source of light, means for rotating bothcylinders while maintaining a selected gear ratio between them, meansfor chopping said light into a series or flashes whereby said photocellgenerates periodically pulses of current, and means for counting saidpulses or current.

4. An apparatus according to claim 3 in which the means for chopping thelight beam into flashes constitutes spaced opaque longitudinal stripeson one of said cylinders. I

EUGENEM. MoNA'I'I.

